Ellipses

Explore focal definition, standard axes, focal distance.

Lesson 3 of 43 parts Practice Flashcards

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Focal Definition

Objective: Explain focal definition and use it to solve or interpret a problem.

An ellipse is a set of points with a constant sum of distances to two foci. A circle is a special ellipse with coincident foci. Yes, with its two foci at the center.

Worked examples

Try itEllipses — practice 1.1: For x²/49+y²/25=1, find the major semiaxis length.Show answer

Step 1: The larger denominator is 49. Step 2: It lies beneath x², so the major axis is horizontal. Step 3: The semiaxis length is √49=7.

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Focal Definition

Objective: Explain focal definition and use it to solve or interpret a problem.

An ellipse is a set of points with a constant sum of distances to two foci. A circle is a special ellipse with coincident foci. Yes, with its two foci at the center.

Worked examples

Try itEllipses — practice 1.1: For x²/49+y²/25=1, find the major semiaxis length.Show answer

Step 1: The larger denominator is 49. Step 2: It lies beneath x², so the major axis is horizontal. Step 3: The semiaxis length is √49=7.

Standard Axes

Objective: Explain standard axes and use it to solve or interpret a problem.

For x²/A²+y²/B²=1 with A>B>0, the major semiaxis is A on the x-axis; exchanging denominators changes orientation. The larger positive squared semiaxis.

Worked examples

Try itEllipses — practice 2.1: For x²/100+y²/64=1, find the major semiaxis length.Show answer

Step 1: The larger denominator is 100. Step 2: It lies beneath x², so the major axis is horizontal. Step 3: The semiaxis length is √100=10.

Focal Distance

Objective: Explain focal distance and use it to solve or interpret a problem.

For major semiaxis A and minor B, focal distance f satisfies f²=A²−B²; it is less than A when B>0. No, because f²=A²−B²<A².

Worked examples

Try itEllipses — practice 3.1: For x²/169+y²/121=1, find focal distance squared.Show answer

Step 1: Use f²=A²−B² for an ellipse. Step 2: Compute 169−121. Step 3: The squared focal distance is 48.

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